Syllabus Honors Geometry



AP Calculus AB Syllabus

Instructor: Paula Ortiz

Phone: 281- 641 6661

Email: paula.ortiz@humble.k12.tx.us

Course Description:

Calculus assimilates many of the concepts of previous mathematics courses, and demonstrates the relevance of the material taught prior to calculus. Calculus AB provides a strong foundation for university-level mathematics and science courses. The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; and Integrals as delineated in the Calculus AB Topic Outline in the AP Calculus Course Description.

First Semester:

Chapter 1: Limits and Their Properties (9 days – one test)

1. • An introduction to limits, including an intuitive understanding of the limit process

2. • Using graphs and tables of data to determine limits

3. • Properties of limits

4. • Algebraic techniques for evaluating limits

5. • Comparing relative magnitudes of functions and their rates of change

6. • Continuity and one-sided limits

7. • Geometric understanding of the graphs of continuous functions

8. • Intermediate Value Theorem

9. • Infinite limits

10. • Using limits to find the asymptotes of a function; discussion of end-behavior of functions

Chapter 2: Differentiation (14 days – three tests)

1. • Zooming-in activity and local linearity

2. • Understanding of the derivative: graphically, numerically, and analytically

3. • Approximating rates of change from graphs and tables of data

4. • The derivative as: the limit of the average rate of change, an instantaneous rate of

5. change, limit of the difference quotient, and the slope of a curve at a point

6. • The meaning of the derivative - translating verbal descriptions into equations

7. • The slope of a curve at a point and the slope of the tangent line to a curve at a point.

1. • The relationship between differentiability and continuity

2. • Functions that have a vertical tangent at a point

3. • Functions that have a point on which there is no tangent

4. • Differentiation rules for basic functions, including power functions and trig functions

5. • Rules of differentiation for sums, differences, products, and quotients

6. • The chain rule

7. • Implicit differentiation

8. • Related rates

9.

Chapter 5: Derivatives of Logarithmic and Exponential Functions (8 days – one test)

• The natural logarithmic function and differentiation

• Exponential functions: differentiation

Chapter 3: Applications of Differentiation (25 days – five tests)

1. • Extrema on an interval and the Extreme Value Theorem

2. • Rolle’s Theorem and the Mean Value Theorem, and their geometric consequences

3. • Increasing and decreasing functions and the First Derivative Test

4. • Concavity and its relationship to the first and second derivatives

5. • Second Derivative Test

6. • Limits at infinity

7. • A summary of curve sketching—using geometric and analytic information as well as

8. calculus to predict the behavior of a function

9. • Relationship between the increasing and decreasing behavior of [pic]and the sign of [pic].

10. • Relationship between the concavity of [pic] and the sign of[pic].

11. • Relating the graphs of [pic]

12. • Optimization including both relative and absolute extrema

13. • Tangent line to a curve and linear approximations

14. • Application problems including position, velocity, acceleration, and rectilinear motion

First Semester Exam (two review days)

Second Semester:

Chapter 4: Integration (18 days—three tests)

1. • Antiderivatives and indefinite integration, including antiderivatives following directly from

2. derivatives of basic functions

3. • Basic properties of the definite integral

4. • Area under a curve

5. • Meaning of the definite integral

6. • Definite integral as a limit of Riemann sums

7. • Riemann sums, including left, right, and midpoint sums

8. • Trapezoidal sums

1. • Use of Riemann sums and trapezoidal sums to approximate definite integrals of

2. functions that are represented analytically, graphically, and by tables of data

3. • Discovery lesson on the First Fundamental Theorem of Calculus

4. • Use of the First Fundamental Theorem to evaluate definite integrals

5. • Use of substitution of variables to evaluate definite integrals

6. • Integration by substitution

7. • Discovery lesson on the Second Fundamental Theorem of Calculus

8. • The Second Fundamental Theorem of Calculus and functions defined by integrals

9. • The Mean Value Theorem for Integrals and the average value of a function

Chapter 5: Logarithmic, Exponential, & Other Transcendental Functions (19 days – four tests)

1. • The natural logarithmic function and integration

2. • Inverse functions

3. • Exponential functions: integration

4. • Bases other than e and applications

5. • Solving separable differential equations

6. • Applications of differential equations in modeling, including exponential growth

7. • Use of slope fields to interpret a differential equation geometrically

8. • Drawing slope fields and solution curves for differential equations

1. • Inverse trig functions and differentiation

2. • Inverse trig functions and integration

Chapter 6: Applications of Integration (8 days – two tests)

1. • The integral as an accumulator of rates of change

2. • Area of a region between two curves

3. • Volume of a solid with known cross sections

4. • Volume of solids of revolution

5. • Applications of integration in problems involving a particle moving along a line,

6. including the use of the definite integral with an initial condition and using the definite

7. integral to find the distance traveled by a particle along a line

Chapter 7: Integration Techniques (4 days – one test)

1. • Review of basic integration rules

2. • Integration by parts

3.

Review for the AP Exam

(3 weeks – three tests)

After the AP Exam:

(9 days – two tests)

Additional topics that are not listed as required topics in the Calculus AB Topic Outline in the AP Calculus Course Description are addressed. Such topics are:

4. • Finding volume using the shell method

5. • L’Hopital’s rule.

6. • Integration by partial fractions

• Work done by a constant force and work done by a variable force.

Projects such as the following are assigned to emphasize and reinforce the calculus used for such:

-Larson’s ideas for drawing characters or objects on the calculator using

calculus statements. Many kinds of stipulations and restrictions can

be put on the number and kind of statements.

-Construct models that illustrate how to find volume using cross-sections.

-Writing calculus programs using Winplot for computation and drawings.

Second Semester Exam (two review days)

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